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Measuring the elongation of vertebrae

September 20, 2013

Let’s take another look at that Giraffatitan cervical. MB.R.2180:C5, from a few days ago:

That’s a pretty elongate vertebra, right? But how elongate, exactly? How can we quantify whether it’s more or less elongate than some other vertebra?

The traditional answer is that we quantify elongation using the elongation index, or EI. This was originally defined by Upchurch (1998:47) as “the length of a vertebral centrum divided by the width across its caudal face”. Measuring from the full-resolution version of the image above, I make that 1779/529 pixels, or 3.36.

But then those doofuses Wedel et al. (2000:346) came along and said:

When discussing vertebral proportions Upchurch (1998) used the term elongation index (EI), defined as the length of the centrum divided by the width of the cotyle. Although they did not suggest a term for the proportion, Wilson & Sereno (1998) used centrum length divided by the height of the cotyle as a character in their analysis. We prefer the latter definition of this proportion, as the height of the cotyle is directly related to the range of motion of the intervertebral joint in the dorsoventral plane. For the purposes of the following discussion, we therefore redefine the EI of Upchurch (1998) as the anteroposterior length of the centrum divided by the midline height of the cotyle.

Since then, the term EI has mostly been used in this redefined sense — but I think we all agree now that it would have been better for Wedel et al to have given a new name to Wilson and Sereno’s ratio rather than apply Upchurch’s name to it.

Aaaanyway, measuring from the image again, I give that vertebra an EI (sensu Wedel et al. 2000) of 1779/334 = 5.33. Which is 58% more elongate than when using the Upchurch definition! This of course follows directly from the cotyle being 58% wider than tall (529/334 pixels).

So one of principal factors determining how elongate a vertebra seems to be is the shape of its cotyle. And that’s troublesome, because the cotyle is particularly subject to crushing — and it’s not unusual for even consecutive vertebrae from the same column to be crushed in opposite directions, giving them (apparently) wildly different EIs.

Here’s an example (though not at all an extreme one): cervicals 4 and 6 of the same specimen, MB.R.2180 (formerly HM SI), as the multi-view photo above:

Measuring from the photos as before, I make the width:height ratio of C4 683/722 pixels = 0.95, and that of C6 1190/820 pixels = 1.45. So these two vertebrae — from the same neck, and with only one other vertebrae coming in between them — differ in preserved cotyle shape by a factor of 1.53.

And by the way, this is one of the best preserved of all sauropod neck series.

Let’s take a look at the canonical well-preserved sauropod neck: the Carnegie Diplodocus, CM 84. Here are the adjacent cervicals 13 and 14, in posterior view, from Hatcher (1901: plate VI):

For C14 (on the left), I get a width:height ratio of 342/245 pixels = 1.40. For C13 (on the right), I get 264/256 pixels = 1.03. So C14 is apparently 35% broader than its immediate predecessor. I absolutely don’t buy that this represents how the vertebrae were in life.

FOR EXTRA CREDIT: what does this tell us about the reliability of computer models that purport to tell us about neck posture and flexibility, based on the preserved shapes of their constituent vertebrae?

So what’s to be done?

The first thing, as always in science, is to be explicit about what statements we’re making. Whenever we report an elongation index, we need to clearly state whether it’s EI sensu Upchurch 1998 or EI sensu Wedel et al. 2000. Since that’s so cumbersome, I’m going propose that we introduce two new abbreviations: EIH (Elongation Index Horizonal), which is Upchurch’s original measure (length over horizontal width of cotyle) and EIV (Elongation Index Vertical), which is Wilson and Sereno’s measure (length over vertical height of cotyle). If we’re careful to report EIH and EIV (or better still both) rather than an unspecified EI, then at least we can avoid comparing apples with oranges.

But I think we can do better, by combining the horizontal and vertical cotyle measurements in some way, and dividing the length by the that composite. This would give us an EIA (Elongation Index Average), which we could reasonably expect to preserve the original cotyle size, and so to give a more reliable indication of “true” elongation.

The question is, how to combine the cotyle width and height? There are two obvious candidates: either take the arithmetic mean (half the sum) or the geometric mean (the square root of the product). Note that for round cotyles, both these methods will give the same result as each other and as EIH and EIV — which is what we want.

Which mean should we use for EIA? to my mind, it depends which is best preserved when a vertebra is crushed. If a 20 cm circular cotyle is crushed vertically to 10cm, does it tend to smoosh outwards to 30 cm (so that 10+30 = the original 20+20) or to 40 cm (so that 10 x 40 = the original 20 x 20)? If the former, then we should use arithmetic mean; if the latter, then geometric mean.

Does anyone know how crushing works in practice? Which of these models most closely approximates reality? Or can we do better than either?

Update (8:48am): thanks for Emanuel Tschopp for pointing out (below) what I should have remembered: that Chure et al.’s (2010) description of Abydosaurus introduces “aEI”, which is the same as one of my proposed definitons of EIA. So we should ignore the last four paragraphs of this post and just use aEI. (Their abbreviation is better, too.)

References

Hatcher, Jonathan B. 1901. Diplodocus (Marsh): its osteology, taxonomy and probable habits, with a restoration of the skeleton. Memoirs of the Carnegie Museum1:1-63 and plates I-XIII.

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12 Responses to “Measuring the elongation of vertebrae”

Chure et al. 2010 in their description of Abydosaurus introduce an average EI, see table 1:
“The elongation of cervical centra has been expressed as centrum length scaled to posterior centrum height (Wilson and Sereno 1998) or to posterior centrum width (Upchurch 1998). Here we scale centrum length to the average of centrum height and width (aEI) to avoid confusing changes in centrum elongation with those in cross-sectional shape as well as to account for deformation. Taxa are listed in order of increasing aEI.”

I do not know enough physics to answer the question “which [mean] is best preserved when a vertebra is crushed?”, but I can make two observations.

1. There are an awful lot of other possible means. One can mention quadratic mean, harmonic mean, and so on. One thing to remember is that every reasonnably behaved function f over the reals gives you a mean: take your two values a and b, compute the arithmetic mean m of f(a) and f(b), and search for m’ such that f(m’)=m. That m’ is a sensible mean. If you take f to be linear, you just get the arithmetic mean; if f is a log, you get the geometric mean; if f is the square function, you get the quadratic mean; and so on. In many cases, choosing the appropriate f boils down to determining which is the linear quantity at stake (i.e., the quantity that can meaningfully be summed and averaged). To take an example, if you want to define the mean square of squares, and are really interested in their area, then you should consider the quadratic mean of their sidelengths. With cubes and volume, it would be the cubic mean, etc.

2. There is a geometrical assumption that makes sense in your case: if the cotyle is crushed mostly in 2D and if there is not much compression, then the area of the cotyle is almost preserved. An ellipsis of great axes a and b has area 2*pi*a*b, so under this hypothesis it is natural to take the geometric mean (because it is a*b that is preserved, not a+b or a^2+b^2 or…).

I did, I just didn’t remember who proposed it. And when I brought this up in email, you assumed I was remembering your independent coinage. :-)

Anyway, I failed by not remembering the aEI authors, BUT I think that your independent invention of aEI is a good sign that it is an intuitive step forward over hEI or vEI (I deliberately reordered the acronyms to bring them into alignment with aEI).

Interesting post, Mike. Benoît has pretty much already said most of what I was going to. If the vertebrae is simply crushed with no shearing along the anterior-posterior axis (eg dorsal portion pushed posteriorly and ventral portion pushed anteriorly) and the material of the bone is not significantly compressed (ie it doesn’t behave like a freshly-baked loaf of bread), then the area of the cotyle (rather, it’s projection onto a plane) will remain constant.

If the planar projection of the cotyle approximates an ellipse it will have a maximum and a minimum “diameter” – the major and minor axes. The area of an ellipse is pi*a*b; where ‘a’ and ‘b’ are respectively half of the major and minor axes (aka semi-major axis and semi-minor axis). This formula is different from that given by Benoît but doesn’t invalidate his method which is to take the geometric mean. Since we’re only interested in ratios we can ignore pi which is a constant. I think that it would also be better (and easier) to measure and use the axes rather than the semi-axes, as this will reduce measurement error and serve to mitigate any distortion cause by possible non-symmetrical crushing (caveat – I know nothing about post-mortem changes in bone morphology).

However, whilst bearing the above caveat in mind, I imagine that things will not always be that straightforward. For instance, if a vertebra is crushed, say dorsoventrally, we naturally assume that it will expand laterally, but might it not also be squished axially, thus making the bone appear to be longer than it was in vivo and artificially increasing the xIE (whichever one we use)?

Shame on me: of course the area is pi*a*b with a, b the semi-axes, as noted by Mark Robintson.

Additional note: instead of measuring two lengthes, it would probably better to measure the area of the a well-chosen section (or, for more commodity, of a well-chosen projection) of the bone. Then, one can normalize by taking the square root, so that the resulting quotient can be compared to others *IE.

The others have mentioned if this… and if that…, which is a great indicator the fact that without thoroughly understanding the taphonomic and material structure conditions before and during deformation, your question cannot be adequately answered. For instance, neither average really works unless you are dealing with nonbiological materials. Most materials, when crushed, change their density as they get compressed, such that a 20×20 object will not compress to the same dimensions as it started, that 20×20 object may become 10×25 or even 10×20. When dealing with things approaching the density of cheetos, such as many dino bones, that can get extreme. Not saying anything here that everyone here doesn’t already know. Of course, this assumes we are just talking about the bones. If buried in mud, a 20×20 object may become 10×40 or 10×50, depending on the characteristics or the mud, infilling amount, and surrounding limitations to flow.
The point to all this is that any of these measurements are going to be misleading if you are working on unreconstructed bones, as you have already surmised.
Heinrich already said all this in a much shorter and concise manner, and I agree with him. I do think looking at the retrodeformation papers to restore symmetry is the way to go before applying the measurements you discuss. They may not restore complete proportions, but it will be a far sight better than trying to do measurements on unreconstructed bones, which are going to be relatively meaningless biologically. There are a number of 3D software programs out there that help with this sort of work. I am sure you are already familiar with some of them.

Without thoroughly understanding the taphonomic and material structure conditions before and during deformation, your question cannot be adequately answered.

I think we all accept that. The issue is that we need some simple, repeatable measure of elongation. At the moment we’re all using EI, which we agree is badly flawed. What I want right now is to replace it with a measurement that is flawed a bit less badly. And I think the candidates are length/((height+width)/2) or length/(sqrt(height*width)). Which to use? One (admittedly cowardly) answer would be to use the former just because Chure et al. did; I wouldn’t be too unhappy with that outcome.

I do think looking at the retrodeformation papers to restore symmetry is the way to go before applying the measurements you discuss. They may not restore complete proportions, but it will be a far sight better than trying to do measurements on unreconstructed bones.

I can’t agree. If the simple process of measuring elongation becomes a complex, irreproducible process involving 3d scans, computers, and (probably) proprietary software, then we will lose all reproducibility, and most likely no-one will bother to do it at all. (I know I wouldn’t — I’ve got better things to do.)

By definition, the most correct process is always the right one. How you define the most correct process is what matters. I think what you mean is the most precise process, which I completely agree, the most precise is not always the most correct process.
The question really comes down to are you producing data that means something or are you just spinning your wheels? Sorry, I just don’t buy the “we know this doesn’t really work, but it’s easy so we’re doing it anyway” type of argument. If you can get a measure that is simple and actually means something, go for it, but if it doesn’t actually produce meaningful results, then you are wasting your time. Measuring deformed bones without taking into account how they are deformed is wasting your time because you aren’t measuring anything biologically relevant.
I do not think it necessarily has to be an overly complicated process to take into account the taphonomy of the bones before taking your measurements. I think it can be done in such a way that the level of precision required to accomplish your task is doable without high tech, complicated software and without having the level of data needed to take into account all relevant factors in an optimal way. The level of precision you need is not overly high and can likely be done with some relatively simple morphometric analyses on a sufficient number of bones to make the reproducibility between bones and between workers sufficient. But it has to be done if you want your measurements to mean anything.
There is plenty of nonproprietary software available for free, you just have to figure out how to use it for your particular purpose. The proprietary ones may have more bells and whistles and may automate things that you may have to think about more in the free versions, but they are available. And if the software is proprietary, why is that a problem, other than making it difficult for everyone to do (while it would be nice for everyone to be able to do whatever experiment they want, that is obviously not an option)? I understand the need to have a transparent system which everyone understands how things are done, but honestly, how many people really understand the calculations involved in multivariate stats and how individual programs perform them? Even more honestly, I think the questions you are asking are part of a morphometric analysis that is a bit beyond a simple ratio anyway.

No — that is precisely what I am saying is not the case. If it was, we’d have had Betamax VCRs instead of VHS, and we’d all have had Unix on our computers since the mid-90s instead of Microsoft crapware. The thing that wins does so by a combination of factors: quality is one, but convenience often trumps it — and price nearly always does.

Measuring EI as it’s currently used takes maybe 30 seconds. Upgrading to the aEI of Chure et al. will up that to maybe 40 seconds. What you’re proposing will increase this to maybe a month of work (coincidentally the amount of time it took us to write the just-released Barosaurus paper). It’s just not going to happen. It’s better to record the aEIs of 1000 vertebrae than your awesome measurement for one.

And your proposed approach of first retrodeforming guarantees that no two workers will get the same result for the same vertebra. It’s a recipe for mixing interpretation in with data. Whereas EI or aEI are solid facts about the artifacts they measure.

[…] a bit of it is derived from three SV-POW! posts: How long was the neck of Diplodocus? (2011), Measuring the elongation of vertebrae (2013) and The Field Museum’s photo-archives tumblr, featuring: airbrushing dorsals (2014). It […]